A registered result in its source context

GRAV.ENTROPIC.COMPRESS.01

Original ledger classification: Numerical [N] (lattice exhibits) + Formal [E] (block identity) + [C] (retyped conjecture) + [O] (AP2 continuum statement open)

Original claim text

THE AP2 COMPRESSION CONJECTURE MADE WELL-POSED (2026-07-13; the last open work package of the v473 entropic-action-bridge round, exercised on an exactly solvable model) -- decides the READING of the conjecture, does NOT prove it; AP2 stays [O]. Background: v473 recorded AP2 as P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2}. On a quasi-free system everything is computable: the boundary restriction is the covariance compression C_A = PCP (v155), the boundary modular data is Peschel's K = ln((1-C_A)/C_A), and for a thermal bulk the Bianconi-shaped bulk identity is exact: f(C_T) = C_T(1-C_T)^{-1} = e^{-h/T}. [N] PURE GAPPED BULK: staggered-mass chain (N=16), ground-state covariance a projection to 1e-15, gap verified -- pure bulk, thermal boundary (Bisognano-Wichmann shape). [N] COMPRESSION IS THERMAL: spec(PCP) strictly inside (0,1), Peschel K finite, entanglement S > 0. [N] THE ORDER OBSTRUCTION (the reading DECIDED): on the pure bulk spec(C) = {0,1}, so f(x) = x/(1-x) is singular on the spectrum -- the literal operator-side reading P f(C) P is ILL-POSED, while the state-side f(PCP) is finite: the compression identity can only be read STATE-SIDE (build Delta_Sigma FROM the compressed relative metric), which matches Bianconi's own construction (her G~ is a LOCAL field, i.e. already the compressed datum) -- AP2 is retyped, not weakened. [E] BLOCK ALGEBRA (symbolic, exact): [C^2]_AA - (C_AA)^2 = B Bt and [C^3]_AA - (C_AA)^3 = A B Bt + B Bt A + B D Bt -- every term carries the cross block TWICE, first order vanishes identically: for ANY analytic f the two readings agree up to SECOND order in the cross-cut correlations; the modular (entanglement) content lives exactly in that quadratic correction. [N] GAP CONTROL at the bounded (covariance) level: the mismatch d = ||[Fermi(h/T)]_AA - Fermi(h_AA/T)|| obeys the quadratic law d <= x^2 (d/x^2 in [0.4, 1.0]) and falls with the gap in the gapped regime while the cross-cut norm falls monotonically; the decoupled cut ([h,P] = 0) agrees EXACTLY -- the two readings converge in the gap-dominated regime, which is where TFPT operates (Delta_eff > 0, v76/v337). [C] AP2 well-posed: state-side reading at the bounded level, mismatch quadratic in cross-cut correlations, gap-suppressed. [O] NOT proven: the continuum seam statement (the Calderon compression of the actual relative metric operator = the actual seam modular operator); no status marker moves anywhere

Original classification detail

AP2 made well-posed [N]/[E]/[C]/[O]: the literal operator-side reading P f(C) P is ill-posed on a pure bulk (f singular on spec {0,1}) => state-side reading forced (matching Bianconi's local construction); block algebra: the mismatch between the two readings is exactly second order in the cross-cut block; gap control: quadratic law d <= x^2, decoupled cut exact, convergence in the gap-dominated regime; AP2 (continuum seam statement) stays [O]

Dependencies as recorded

GRAV.ENTROPIC.ACTION.01; GRAV.ENTROPIC.SCALARON.01; v76; v155; v337; v473